doi:10.1016/j.jcta.2005.01.012 
Copyright © 2005 Elsevier Inc. All rights reserved.
Optimal tristance anticodes in certain graphs
aTechnion — Israel Institute of Technology, Department of Computer Science, Technion City, Haifa 32000, Israel
bDepartment of Computer Science, University of California San Diego, La Jolla, CA 92093, USA
cDepartment of Electrical Engineering, University of California San Diego, La Jolla, CA 92093, USA
dDepartment of Mathematics, University of California San Diego, La Jolla, CA 92093, USA
Received 6 June 2004.
Communicated by Vera Pless.
Available online 13 May 2005.
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Abstract
For 
, the tristance d3(z1,z2,z3) is a generalization of the L1-distance on 
to a quantity that reflects the relative dispersion of three points rather than two. A tristance anticode 
of diameter d is a subset of 
with the property that d3(z1,z2,z3)
d for all 
. An anticode is optimal if it has the largest possible cardinality for its diameter d. We determine the cardinality and completely classify the optimal tristance anticodes in 
for all diameters d
1. We then generalize this result to two related distance models: a different distance structure on 
where d(z1,z2)=1 if z1,z2 are adjacent either horizontally, vertically, or diagonally, and the distance structure obtained when 
is replaced by the hexagonal lattice A2. We also investigate optimal tristance anticodes in 
and optimal quadristance anticodes in 
, and provide bounds on their cardinality. We conclude with a brief discussion of the applications of our results to multi-dimensional interleaving schemes and to connectivity loci in the game of Go.
Keywords: Anticodes; Grid graph; L1-distance; Multidimensional interleaving; Tristance
Abstract
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